c)Using the Mean Value Theorem show that f(x)=0 has only one root in the interval (0,1)
We shall prove this by contradiction. (See: Proof By Contradiction).
Assume that f(x) has 2 roots, c1 and c2 in the given interval.
f(c1) = f(c2) = 0
By the mean value theorem we have:
f'(c).(c2 - c1) = f(c2) - f(c1)
f'(c) = 0
But in part a it was shown that f'(c) > 0 in the range (a,b)
We have a contradiction, and thus the original assumption that there are two roots, has been disproven.

There is actually a very simple way to understand this physically. If a function is everywhere differentiable then the only way its graph can turn is if its derivative becomes zero and then changes sign.
This means that if a differentiable function crosses the x-axis once then unless its derivative becomes zero and changes sign it cannot turn back for another crossing. If the derivative is always positive the function keeps increasing and if the derivative is always negative then the function keeps decreasing. Thus if the derivative maintains its sign, the function cannot have two roots.